Optimal. Leaf size=76 \[ -\frac{2 (c-a c x)^{3/2}}{3 a c}-\frac{4 \sqrt{c-a c x}}{a}+\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{a} \]
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Rubi [A] time = 0.102095, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6167, 6130, 21, 50, 63, 206} \[ -\frac{2 (c-a c x)^{3/2}}{3 a c}-\frac{4 \sqrt{c-a c x}}{a}+\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6130
Rule 21
Rule 50
Rule 63
Rule 206
Rubi steps
\begin{align*} \int e^{-2 \coth ^{-1}(a x)} \sqrt{c-a c x} \, dx &=-\int e^{-2 \tanh ^{-1}(a x)} \sqrt{c-a c x} \, dx\\ &=-\int \frac{(1-a x) \sqrt{c-a c x}}{1+a x} \, dx\\ &=-\frac{\int \frac{(c-a c x)^{3/2}}{1+a x} \, dx}{c}\\ &=-\frac{2 (c-a c x)^{3/2}}{3 a c}-2 \int \frac{\sqrt{c-a c x}}{1+a x} \, dx\\ &=-\frac{4 \sqrt{c-a c x}}{a}-\frac{2 (c-a c x)^{3/2}}{3 a c}-(4 c) \int \frac{1}{(1+a x) \sqrt{c-a c x}} \, dx\\ &=-\frac{4 \sqrt{c-a c x}}{a}-\frac{2 (c-a c x)^{3/2}}{3 a c}+\frac{8 \operatorname{Subst}\left (\int \frac{1}{2-\frac{x^2}{c}} \, dx,x,\sqrt{c-a c x}\right )}{a}\\ &=-\frac{4 \sqrt{c-a c x}}{a}-\frac{2 (c-a c x)^{3/2}}{3 a c}+\frac{4 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0317838, size = 61, normalized size = 0.8 \[ \frac{2 (a x-7) \sqrt{c-a c x}+12 \sqrt{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c-a c x}}{\sqrt{2} \sqrt{c}}\right )}{3 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 59, normalized size = 0.8 \begin{align*} -2\,{\frac{1}{ac} \left ( 1/3\, \left ( -acx+c \right ) ^{3/2}+2\,c\sqrt{-acx+c}-2\,{c}^{3/2}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{-acx+c}\sqrt{2}}{\sqrt{c}}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84861, size = 313, normalized size = 4.12 \begin{align*} \left [\frac{2 \,{\left (3 \, \sqrt{2} \sqrt{c} \log \left (\frac{a c x - 2 \, \sqrt{2} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a x + 1}\right ) + \sqrt{-a c x + c}{\left (a x - 7\right )}\right )}}{3 \, a}, -\frac{2 \,{\left (6 \, \sqrt{2} \sqrt{-c} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{-c}}{2 \, c}\right ) - \sqrt{-a c x + c}{\left (a x - 7\right )}\right )}}{3 \, a}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.71487, size = 73, normalized size = 0.96 \begin{align*} - \frac{2 \left (\frac{2 \sqrt{2} c^{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{- a c x + c}}{2 \sqrt{- c}} \right )}}{\sqrt{- c}} + 2 c \sqrt{- a c x + c} + \frac{\left (- a c x + c\right )^{\frac{3}{2}}}{3}\right )}{a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12162, size = 104, normalized size = 1.37 \begin{align*} -\frac{4 \, \sqrt{2} c \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c}}{2 \, \sqrt{-c}}\right )}{a \sqrt{-c}} - \frac{2 \,{\left ({\left (-a c x + c\right )}^{\frac{3}{2}} a^{2} c^{2} + 6 \, \sqrt{-a c x + c} a^{2} c^{3}\right )}}{3 \, a^{3} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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